The Triple Intergral of the given expression is [tex]4 \pi \times ln (\frac{a}{b})[/tex]
[tex]\int\limits \int\limits \int\limits_w \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}[/tex]
[tex]\text{Where } x^2 + y^2 + z^2 = a^2 \text{ and } x^2 + y^2 + z^2 = b^2, 0 < b < a[/tex]
Changing to Spherical Coordinates
[tex]b \leq \rho \leq a, 0 \leq \theta \leq 2 \pi, 0 \leq \phi \leq \pi[/tex]
[tex]\therefore x^2 + y^2 + z^2 = \rho^2; dx dy dz = \rho^2 sin \phi .d\phi .d\theta. d\rho[/tex]
[tex]\implies \int\limits^a_b \int\limits^{2\pi}_{0} \int\limits^{\pi}_{0} \frac{1}{\rho^3} \rho^2 sin \phi .d\phi .d\theta. d\rho[/tex]
[tex]=\int\limits^a_b {\frac{1}{\rho}} \, d\rho \int\limits^0_{2\pi} d\theta \int\limits^{\pi}_0 \sin{\phi} \, d\phi[/tex]
[tex]= [ln \rho]^{a}_{b} [\theta]^{2\pi}_{0} [-\cos \phi]^{\pi}_{0}[/tex]
[tex]= [\ln a - \ln b] \times [2\pi] \times [- \cos \pi - (-\cos 0)][/tex]
[tex]= 2\pi \times 2 \times (\ln a - \ln b)[/tex]
[tex]= 4 \pi \, \ln{\frac{a}{b}}[/tex]
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Susan plans to attend college in 3 years. She will need $ 17,400 for the first-year cost. Her uncle gave her a special gift of $12,000 to use toward the cost. How much must Susan save each month to have enough for the first-year cost?
Answer:
$150 per month
Step-by-step explanation:
Susan plans to attend college in 3 years. She will need $ 17,400 for the first-year cost. Her uncle gave her a special gift of $12,000 to use toward the cost. How much must Susan save each month to have enough for the first-year cost?
17,400 - 12,000 = $5,400 Susan needs to pay in 3 years.
3 years = 36 months
$5,400/36 = $150 per month
Let a,b,c be positive real numbers. Prove the inequality
[tex]\displaystyle\\\bf\frac{(a+b)^2}{c} +\frac{c^2}{a} \geq 4b[/tex]
Answer: proved
Step-by-step explanation:
Prove the inequality:
[tex]\displaystyle\\\frac{(a+b)^2}{c} +\frac{c^2}{a} \geq 4b\ \ \ \ \ a > 0\ \ \ \ \ b > 0\ \ \ \ \ c > 0\\[/tex]
Multiply both parts of the inequality by ac (a>0, c>0):
[tex]a(a+b)^2+c*c^2\geq 4abc[/tex]
Simplify the left side of the inequality using the Cauchy inequality:
[tex]\boxed {t+v\geq 2\sqrt{tv} }[/tex]
[tex]a(a+b)^2+c(c^2)\geq 2\sqrt{a(a+b)^2c(c^2)} \\\\a(a+b)^2+c(c^2)\geq 2(a+b)c\sqrt{ac} \\\\a(a+b)^2+c(c^2)\geq 2(ac+bc)\sqrt{ac}\ \ \ \ \ (1)[/tex]
Simplify the right side of the inequality using the Cauchy inequality:
[tex]ac+bc\geq 2\sqrt{acbc} \\\\ac+bc\geq 2\sqrt{abc^2} \\\\ac+bc\geq 2c\sqrt{ab}[/tex]\ \ \ \ \ (2)
Substitute expression (2) into expression (1):
[tex]a(a+b)^2+c(c^2)\geq 2*2c\sqrt{ab} \sqrt{ab} \\\\a(a+b)^2+c(c^2)\geq4abc[/tex]
Hence,
[tex]\displaystyle\\\frac{(a+b)^2}{c} +\frac{c^2}{a} \geq 4b[/tex]
Graph the line with -intercept -4 and slope 6
Can you mark the points
The graph of the given line with y-intercept -4 and slope 6 is given below.
What is slope intercept form?
One of the most popular ways to represent the equation of a line is in the slope intercept form of a straight line. When the slope of the straight line and the y-intercept are known, the slope-intercept formula can be used to determine the equation of a line ( the y-coordinate of the point where the line intersects the y-axis).
For a straight line with slope "m" and y-intercept "b," the slope-intercept form equation is:
y = mx + b.
Given, y-intercept (b) = -4
slope (m) = 6
Then, the equation of the graph will be
y = 6x - 4
The graph of the equation with points is given below:
Hence, this is the required graph of the slope-intercept form line.
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a norman window consists of a rectangle surmounted by a semicircle. if the perimeter of a norman window is to be 32 feet, determine what should be the radius of the semicircle and the height of the rectangle such that the window will admit the most light.?4.49 ft.
Therefore, the height of the rectangle and the radius of the semicircle should be 8 feet each in order for the Norman window to admit the most light.
What is the perimeter of a semi circle?The distance along the line defining a half circle's boundary is known as the semi circle's perimeter. It is made up of the diameter of the semi circle, which is a straight line that passes through the center of the circle, and the semi circle's curving arc. For example, he radius of the semicircle and the height of the rectangle such that the window will admit the most light. A semi circle's diameter must be determined before calculating the circumference. This may be accomplished by determining the radius of the semicircle, which is the separation between any two points on the circle and the circle's center. Simply said, the diameter is twice the radius.
How to solve?
If the perimeter of the Norman window is 32 feet, then each side of the rectangle must be 8 feet long. ( using 2(l+b))
The radius of the semicircle must also be 8 feet, as the perimeter of a semicircle is equal to 2 * pi * radius.
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he table of values represents a quadratic function f(x).
x f(x)
−7 14
−6 7
−5 2
−4 −1
−3 −2
−2 −1
−1 2
0 7
1 14
What is the equation of f(x)?
f(x) = (x − 4)2 − 1
f(x) = (x − 3)2 − 2
f(x) = (x + 3)2 − 2
f(x) = (x + 4)2 − 1
The equation of the quadratic function is (c) f(x) = a(x + 3)² - 2
How to determine the equation of the quadratic function?From the question, we have the table of values that can be used in our computation:
A quadratic equation is represented as
f(x) = a(x - h)² + k
Where
Vertex = (h, k)
From the graph, we have the vertex to be
(h, k) = (-3, -2)
Substitute (h, k) = (-3, -2) in f(x) = a(x - h)² + k
So, we have
f(x) = a(x + 3)² - 2
Also, from the graph, we have the point (0, 7)
This means that
a(0 + 3)² - 2 = 7
So, we have
9a = 9
Divide both sides by 9
a = 1
Substitute a = 1 in f(x) = a(x + 3)² - 2
f(x) = a(x + 3)² - 2
Hence, the equation is f(x) = a(x + 3)² - 2
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Answer:
hey um im back....
Step-by-step explanation: i was in the ER from being abused and you did not lose me don't worry i love you i was moving place to place so i did not have my phone i wish i could undo this all but im ok now love
michelle
3. List all the factors of the number 30. (1 point) 1, 2, 3, 5, 6, 10, 15, 30 2, 3, 4, 10, 20, 30, 40 1, 2, 4, 5, 10, 15, 30 1, 2, 4, 5, 8, 10, 20, 40
Answer:
1,2,3,5,6,10,15,30
Step-by-step explanation:
1*30
2*15
3*10
5*6
Jared draws a rectangle. Explain how
to find the area using the Distributive
9 ft length
5 ft height
The populations of 2 cities grow according to the exponential functions. Pi(t) = 120 e 0.011 t P2(t)= 125 e 0.007 t Where, P₁ and P2 are the populations (in thousands) of cities A and B respectively. t is the time in years such that t is positive and t = 0 corresponds to the year 2004. When were the populations of the two cities equal and what were their populations?
Hello,
I hope you and your family are doing well!
To find when the populations of the two cities are equal, we can set P1(t) = P2(t) and solve for t:
120 e 0.011 t = 125 e 0.007 t
Dividing both sides by e 0.007 t:
120 = 125 * (e 0.011 t / e 0.007 t)
Using the property that e^(a+b) = e^a * e^b, we have:
120 = 125 * e^(0.011 t - 0.007 t)
120 = 125 * e^(0.004 t)
Dividing both sides by 125:
1.2 = e^(0.004 t)
Taking the natural logarithm of both sides:
ln 1.2 = ln e^(0.004 t)
ln 1.2 = 0.004 t
t = ln 1.2 / 0.004
t = approximately 8.44 years
Therefore, the populations of the two cities were equal approximately 8.44 years after 2004, or in the year 2012. To find the population of the two cities at this point in time, we can substitute t = 8.44 into either of the exponential functions:
P1(8.44) = 120 e 0.011 * 8.44 = approximately 123.88 thousand
P2(8.44) = 125 e 0.007 * 8.44 = approximately 123.88 thousand
Therefore, the populations of the two cities were equal to approximately 123.88 thousand when they were equal.
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Happy Holidays!
Use the graph to answer the question. Where was the object after 2 seconds?
A. 6m
B. 5m
C. 14m
D. 4m
Answer: A. 6m
Step-by-step explanation: the graph shows at the x axis that it is time. so, go to two seconds and then go up to where the line is. it goes up to 6m so that is the answer.
what is the value of x
The value of x is [tex]x=\frac{268}{3}[/tex] = 89.33 degree.
How do you find value of x?
The algebraic expression should often take one of the following forms: addition, subtraction, multiplication, or division. Bring the variable to the left and the remaining values to the right to determine the value of x. To determine the outcome, simplify the values. The letter "x" is frequently used in mathematics to denote an unknowable amount or variable. Similar to how X-rays confounded their discoverer and Malcolm X adopted the sign to stand for the lost name of his African ancestry, x also denotes the unknown in English.
x+44+x+3+x+45=360
Group like terms
x+x+x+44+3+45=360
Add similar elements: x+x+x=3 x
3 x+44+3+45=360
Add the numbers: 44+3+45=92
3 x+92=360
Move 92 to the right side
3 x=268
Divide both sides by 3
[tex]\frac{3 x}{3}=\frac{268}{3}[/tex]
Simplify
[tex]x=\frac{268}{3}[/tex]
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The measurement of the radius of the end of a log is found to be 8 inches, with a possible error of 1/8 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.
By using the differentials to approximate the possible propagated error in computing the area of the end of the log is ±2π
The measurement of the radius of the end of a log = 8 inches
The possible error of radius = 1/8 inches
The area of cross section of the end of the log = πr^2
A = πr^2
dA / dr = 2πr
dA = 2πr(dr)
Here the possible error of radius = 1/8 inches
dr = 1/8 inches
Substitute the values in the equation
dA = 2π × 8 × (± 1/8)
= ±2π
Therefore, the possible propagated error in computing the area is ±2π
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If a snowball melts so that its surface area decreases at a rate of 4 cm²/min, find the rate at which the diameter
decreases when the diameter is 39 cm.
The surface area of a snowball can be represented by the formula:
A = 4πr^2
where A is the surface area, r is the radius of the snowball, and π is a constant approximately equal to 3.14.
If the surface area of the snowball decreases at a rate of 4 cm²/min, we can represent this as a derivative:
dA/dt = -4
where dA/dt is the rate of change of the surface area with respect to time, and t is the time in minutes.
The radius of the snowball is equal to half the diameter, so we can represent the radius as:
r = d/2
where d is the diameter of the snowball.
Substituting this expression for r into the formula for the surface area and differentiating with respect to time, we get:
dA/dt = 8π(d/2)dr/dt
Setting this expression equal to -4 and solving for dr/dt, we get:
dr/dt = -4 / (8π)
= -0.5 / π
When the diameter of the snowball is 39 cm, the rate at which the diameter decreases is:
dr/dt = -0.5 / π
= -0.15915494309189533576888376337251
Therefore, the rate at which the diameter of the snowball decreases when the diameter is 39 cm is approximately -0.159 cm/min.
I hope this helps
what is the answer to this 6m+8=4(17+m)
Answer:
m=30
Step-by-step explanation:
7. You need to find the height of a nearby tree. Instead of climbing the tree, you decide to
use the length of your shadow and the shadow of the tree. Use the image below to find
the height (h) of the tree to the nearest foot.
The height of a nearby tree is 15ft
What is Application of trigonometry ?
Trigonometry is the study of lengths, heights, and angles through computations involving triangles. There are many applications for trigonometry and its functions in daily life. For instance, it is employed in astronomy to gauge the distances between nearby stars, in geography to gauge the separations between landmarks, and in the satellite navigation system.
According to question
let
The height of tree = h
Angle elevation = α
Given
Length of shadow of tree = 20 ft
height of man = 6 ft
Length of shadow of man = 8 ft
We know that
tanθ = Perpendicular/Base
therefore:
tanα = 6/8 = h/20
⇒ 6*20 = 8h
⇒ 120 = 8h
⇒ h = 120/8
⇒ h = 15
hence, the height of a nearby tree is 15 ft.
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The function f(x)=(−x+4)(2x+2) is written in factored form.
Drag statements into the table to show what each factor tells you about the graph of the function. Imagine math
The intersection of the curve with the x-axis will be at (−1, 0) and (4, 0) for each factor.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
The function written in factored form is given below.
f(x) = (−x + 4) (2x + 2)
For the factor (−x + 4), we have
−x + 4 = 0
x = 4
For the factor (2x + 2), we have
2x + 2 = 0
2x = −2
x = −1
The intersection of the curve with the x-axis will be at (−1, 0) and (4, 0) for each factor.
The graph is given below.
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A movie theater has a seating capacity of 367. The theater charges $5.00 for children, $7.00 for
students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket
sales was $2660 on a sold out night, how many children, students, and adults attended?
Number of children adults and students who attended the theatre is 182 , 91 , 94 respectively
Let Number of Children in theatre = c
Let Number of Adults in theatre = a
Let Number of Students in Theatre =s
Total money earned by selling the tickets = $2660
According to the given information
c + s + a = 367
a = c/2
5c + 7s + 12a = 2660
5c + 5s + 5a = 5(367)= 1835
Using elimination and substitution method
subtract
2s +7a = 2660-1835 = 825
2s +7(c/2) = 2s +7c/2 = 825
4s +7c = 1650
c + s + c/2 = 367
2c + 2s + c =734
2s + 3 c =734
4s + 6c=1468
4s + 7c =1650
c = 182 children
a= 91 adults
s = 367-273 = 94 students
Number of children adults and students who attended the theatre is 182 , 91 , 94 respectively
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5. Determine the coordinates of point Y so that JY = (2/3)JK.
The coordinates of y so that JY = (2/3)JK is Y = 2/3k
What are coordinates?Coordinates are a pair of numbers (Cartesian coordinates), or sporadically a letter and a number, that identify a certain place on a grid, also referred to as a coordinate plane.
The x axis (horizontal) and y axis are the two axes that make up a coordinate plane (vertical)
How to find the coordinates of YExamining the expression given JY = (2/3)JK, it can be seen that
at the left part of the equation Y is present while at the second part of the equation Y is no replaced by 2/3k
Hence the 2/3k is equal to Y, and this represents the points that defines Y
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Profit margin ratio 294\3800
Step-by-step explanation:
12.93 is the answer of this question.
In a certain chemical, the ratio of zinc to copper is 4 to 17 . A jar of the chemical contains 561 of copper. How many grams of zinc does it contain?
Answer:
7,293 if I am correct.
Hope this helps!
What inverse trig function would be used to solve for angle Z?
The inverse sine trig function would be used to solve for angle Z .
What is meant by inverse sine ?In the right angle triangle the ratio of the opposite to the hypotenuse of any acute <A is called the sine of < A
The inverse of the sine function, also known as the arcsine function, returns the angle's value when the sine function's opposite side and hypotenuse ratio are equal. It produces the angle's value.
The opposite of the sine function is the inverse sine function, also known as arcsine. Due to the fact that the sine of an angle (or sine function) is equal to the ratio of the opposite side to the hypotenuse, the sine inverse of this ratio will provide the angle's measurement.
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50 POINTS BABY!! PLEASE HELP ASAP
Answer:
I think 50
Step-by-step explanation:
Find how long it takes $2500 to double if it is invested at 6% interest compounded semiannually. Use the formula A=P(1+r/n) to solve the compound interest problem.
It will take approximately years.
(Do not round until the final answer. Then round to the nearest tenth as needed.)
After 11.71 years the amount will double to the principal amount
What is Compound interest:Compound interest refers to the interest added to a loan or deposit that depends on both the principal amount and compound interest rate and the number of compoundings in a year.
The formula for the amount in compound interest is given by
Amount, A = P(1+r/n)^ntwhere P = Principal amount
r = Rate of interest
n = Number of compounds
t = Time period
From the given data
Principal amount, p = $ 2500
Rate of interest, r = 6% = 6/100 = 0.06
Number of compounds, n = 2 [ ∵ 12/6 = 2 ]
Here we need to find the time period
Let's assume that after 't' years the amount will be doubled i.e $ 5000
From the given problem,
Amount, A = 2500(1+0.06/2)^2t
= 2500(1+0.06/2)^2t
= 2500(1 + 0.03)^2t
= 2500 (1.03)^2t
As we assumed after 't' years the amount is $ 5000
=> 2500 (1.03)^2t = 5000
=> (1.03)^2t = 5000/2500
=> (1.03)^2t = 2
Apply to log on both sides
=> log (1.3)^t = log 2
=> 2t log (1.3) = log 2
=> 2t = log 2/ log(1.03)
=> 2t = 23.4497722
[ use calculator for log (1.03) and log 2 values]
=> t = 11.7248861
=> t = 11.71 ≅ 11 years
Therefore,
After 11.71 years the amount will double to the principal amount
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Using complete sentences, explain thoroughly what is meant by the given formula:
a 1 =
: 10
an = an-1+4
The meaning of the given formula is that the first term (a₁) is equal to the nth term (aₙ) of a sequence multiplied by 10 and it is also equal to the preceding nth term (aₙ₋₁) of a sequence plus four (4).
What is an arithmetic sequence?In Mathematics, an arithmetic sequence can be defined as a series of real and natural numbers in which each term differs from the preceding term by a constant numerical quantity.
How to calculate the nth term of an arithmetic sequence?Mathematically, the nth term of an arithmetic sequence can be calculated by using this expression:
aₙ = a₁ + (n - 1)d
Where:
d represents the common difference.a₁ represents the first term of an arithmetic sequence.n represents the total number of terms.Based on the information provided, we can logically deduce that the first term (a₁) of the arithmetic sequence can either be calculated by multiplying the nth term (aₙ) by ten 10 or adding four (4) to the preceding nth term (aₙ₋₁) of the arithmetic sequence.
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For the following, find the arc length:
(Use \large \pi=3.14 and round your final answer to the hundredths.)
x=_________
Answer:
x = 10.47
Step-by-step explanation:
circumference = 2 * 10 * π = 20π
centre angle of a circle (in radiant) = 2π
x : 20π = π/3 : 2π
x = (20π * π/3)/ 2π
x = 10 * π/3
x = 10π/3
x = 10.466667
x = 10.47
Consider the following statement: Let f be a function, such that for all a and b in the domain of f, f(a)=f(b) implies a=b. Then the function f has an inverse. Is this statement true or false?
True, the given function f has an inverse.
A function that can reverse into another function is known as an inverse function or anti-function. In other words, the inverse of a function "f" will take y to x if any function "f" takes x to y. Inverse functions f and g result in f(x) = y if and only if g(y) = x.
Under the given condition, we can define the function g on the set of the images {b | b = f(a), a belongs to A} in a way
g(b) = a, if f(a) = b.
This function is defined correctly on this set and is the inverse function to f.
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Your family plans to start a small business in your neighborhood. Your father borrows
$10, 000 from the bank at an annual interest rate of 8% rate for 36 months. What is the
amount of interest he will pay on this loan?
The amount of interest he will pay on this loan$12,400.
What is interest?
The fee you pay to borrow money or the fee you charge to lend money is called interest. The most common way to represent interest is as an annual percentage of the loan amount. The interest rate on the loan is known as this percentage.
Principal = $10,000
Rate = 8%
Time = 36 months, 1 year - 12 months
36 months - 36/12= 3 years
S.I. = PRT/100
10,000*8*3/100
240,000/100 = $2,400
A = P+I = 10,000+2,400= $12,400
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Graphing the line with.....
Can you also mark the two points on the graph.
The graph of the linear equation y = (3/2)*x - 7
Can be seen in the image at the end.
How to graph the linear given equation?A general linear equation is written as:
y = a*x + b
Where a is the slope and b is the y-intercept.
Here we know that the slope is -2 and the y-intercept is -4, then the linear equation is:
y = -2x - 4
To graph that line we need to find two points on it, to get the points we need to replace the value of x.
if x = 0
y = -2x - 4
y = -4
So we have the point (0, -4)
And if x = 1:
y = -2*1 - 4
y = -6
So we have the point (1, -6)
Now we just need to graph these two points and connect them with a line.
You can see the graph below.
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Which graph shows a proportional relationship between the number of hours of renting a lawn mower and the total amount spent to rent the lawn mower? PLEASE HELP! ASAP
During the school fundraiser 4/6 of the products sold were flavored popcorn. Of the flavored popcorn, 1/2 who was caramel flavored. What fraction of the fundraiser products is caramel flavored popcorn?
Answer:
3/10
Step-by-step explanation:
please mark me brainliest if this is right
help please easy math proportional relationships, due now please help needed
By knowing that the cost equation must be a proportional relation, we will find that the equation is:
y = $0.64*x
Which equation can be used to find the cost of x pounds?
The cost equation is proportional, if we define y as the cost and x as the number of pounds, the equation is something like:
y = k*x
We know that for (5 + 1/2) pounds the cost is $3.52, then we can replace these values in the equation above to get.
$3.52 = k*(5 + 1/2)
$3.52/(5 + 1/2) = k = $0.64
Then the cost equation for x pounds of potatoes is:
y = $0.64*x
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